Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite Elements

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Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite

Elements

Pascal Azerad, Jean-Luc Guermond, Bojan Popov

To cite this version:

Pascal Azerad, Jean-Luc Guermond, Bojan Popov. Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite Elements. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2017, 55 (6), pp.3203 - 3224. �10.1137/17M1122463�.

�hal-01815500�

WELL-BALANCED SECOND-ORDER APPROXIMATION OF THE SHALLOW WATER EQUATION WITH CONTINUOUS FINITE

ELEMENTS

^∗

PASCAL AZERAD^†, JEAN-LUC GUERMOND^‡, AND BOJAN POPOV^‡

Abstract. This paper investigates a first-order and a second-order approximation technique for the shallow water equation with topography using continuous finite elements. Both methods are explicit in time and are shown to be well-balanced. The first-order method is invariant domain preserving and satisfies local entropy inequalities when the bottom is flat. Both methods are positivity preserving. Both techniques are parameter free, work well in the presence of dry states, and can be made high order in time by using strong stability preserving time stepping algorithms.

Key words.shallow water, well-balanced approximation, invariant domain, second-order method, finite element method, positivity preserving

AMS subject classifications. 65M08, 65M60, 65M12, 35L50, 35L65, 76M10 DOI. 10.1137/17M1122463

1. Introduction. The objective of this paper is to develop an invariant do- main preserving well-balanced approximation of the shallow water equation with bathymetry using continuous finite elements. There are many finite volume and dis- continuous Galerkin (DG) techniques available in the literature that can solve this problem efficiently up to second and higher order in space. Examples of schemes that are well balanced at rest and robust in the presence of dry states can be found, for example, in Audusse and Bristeau [1], Audusse et al. [2], Bollermann, Noelle, and Luk´ aˇ cov´ a-Medvidov´ a [6], Gallardo, Par´ es, and Castro [14], Kurganov and Petrova [23], Perthame and Simeoni [27], Ricchiuto and Bollermann [28]. We refer the reader to the book of Bouchut [7] for a review on this topic, to the paper of Xing and Shu [32] for a survey on finite volume and DG methods, and to the paper [23] for a survey of central-upwind schemes. However, to the best of our knowledge, these types of approximations are not developed in the context of continuous finite elements. Or we should say that no robust continuous finite element technique is yet available in the literature that guarantees second-order accuracy, works properly in every regime (subcritical, transcritical, transcritical with hydraulic jumps, wet, and dry regions) and is well-balanced at rest. We propose such a method in the present paper. Two variants of the method are discussed: one variant is first-order accurate in space, positivity preserving, and preserves every convex invariant domain of the system in the absence of bathymetry; the other variant is second-order accurate in space and positivity preserving. Both variants are explicit in time and use continuous finite elements on unstructured meshes.

∗Received by the editors March 24, 2017; accepted for publication (in revised form) September 11, 2017; published electronically December 19, 2017.

http://www.siam.org/journals/sinum/55-6/M112246.html

Funding: The work of the authors was supported in part by the National Science Foundation grants DMS-1619892 and DMS-1620058, by the Air Force Office of Scientific Research, USAF, under grant/contract FA9550-15-1-0257, and by the Army Research Office under grant/contract W911NF- 15-1-0517.

†Institut Montpellierain Alexander Grothendieck, UMR 5149, Universit´e de Montpellier, 34095 Montpellier, France (azerad@math.univ-montp2.fr).

‡Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843 (guermond@math.tamu.edu, popov@math.tamu.edu).

3203

The first building block of the method consists of using the methodology intro- duced in Guermond and Popov [16]. The second building block consists of making the schemes well-balanced with respect to rest states by using the so-called hydrostatic reconstruction from [2, section 2.1] and variations thereof. The technique from [16] is a loose extension of Lax’s scheme [24, p. 163] to continuous finite elements; it solves general hyperbolic systems in any space dimension using forward Euler time step- ping and continuous finite elements on nonuniform grids. The artificial dissipation is defined so that any convex invariant set containing the initial data is an invariant domain for the method. The solution thus constructed satisfies a discrete entropy in- equality for every admissible entropy of the system. The accuracy in space is formally first order and the accuracy in time can be made high order by using strong stability preserving Runge–Kutta time stepping. Some ideas of the method are rooted in the work of Hoff [20, 21], and Frid [13]. The method is made second order and positivity preserving by using techniques introduced in Guermond and Popov [17].

The paper is organized as follows. The model problem and the finite element setting are introduced in section 2. The first-order variant of the method is described in section 3. The main results of this section are Propositions 3.9 and 3.11. The second-order variant of the method is described in section 4. The key results of this section are Propositions 4.2 and 4.4. The performances of the algorithms introduced in the paper are numerically illustrated in section 5 on standard benchmark problems.

2. Preliminaries. In this section we introduce the model problem, the finite element setting, and we define (recall) the concept of well-balancing at rest.

2.1. The model problem. Let D be a polygonal domain in R

^d

with d ∈ {1, 2}, occupied by a body of water evolving in time under the action of gravity. Assuming that the deformations of the free surface are small compared to the water elevation and the bottom topography z varies slowly, the problem can be well represented by Saint- Venant’s shallow water model. This model describes the time and space evolution of the water height h and flow rate, or discharge, q in the direction parallel to the bottom. Using u = (h, q)

^T

as a dependent variable the model is as follows:

∂

_t

u + ∇·f (u) + b(u, ∇z) = 0, x ∈ D, t ∈ R

+

, (2.1)

f (u) :=

q

^T

1

h

q⊗q +

¹₂

gh

²

I

d

∈ R

^(1+d)×d

, b(u, ∇z) :=

0 gh∇z

. (2.2)

The quantity q is related to the horizontal component of the water velocity v by q = vh. The function z : D 3 x 7→ z(x) ∈ R is the given topography.

We assume that either the boundary conditions are periodic or the initial data u

0

and the bottom topography are constant outside a compact set in D and the solution to (2.1) is constant outside this compact set over some time interval [0, T ].

2.2. The finite element space. We approximate the solution of (2.2) with continuous finite elements. Let (T

h

)

_h>0

be a shape-regular family of matching meshes.

(Here we slightly abuse notation by denoting the mesh size by h. For instance we

are going to denote by h

_h

the finite element approximation of the water height.) The

elements in T

_h

are assumed to be generated from a finite number of reference elements

denoted { K b

r

}

_1≤r≤$

. For example, the mesh T

h

could be composed of a combination

of triangles and quadrangles ($ = 2 in this case). Given a set of reference finite

elements in the sense of Ciarlet {( K b

r

, P b

r

, Σ b

r

)}

_1≤r≤$

(the index r ∈ {1:$} is omitted

in the rest of the paper to simplify the notation) we introduce the finite element space (2.3) P(T

h

) := n

v ∈ C

⁰

(D; R ) | v

_|K

◦T

K

∈ P , b ∀K ∈ T

h

o ,

where for any K ∈ T

h

, T

K

: K b → K is the geometric bijective transformation that maps the reference element K b to the current element K. We do not assume that T

K

is affine. The exact nature of the degrees of freedom in Σ b

r

is not essential, but the reader who is not familiar with finite elements can think of Lagrange elements or Bernstein elements. The reference space P b is assumed to be composed of scalar- valued functions (these are polynomials usually). The reference shape functions are denoted {b θ

_i

}

_i∈{1:nsh}

; recall that they form a basis of P b . We assume that the basis {b θ

i

}

_i∈{1:nsh}

has the partition of unity property: P

i∈{1:nsh}

b θ

i

( x) = 1 for all b x b ∈ K. b The approximation in space of u in (2.2) will be done in P (T

h

) := [P (T

h

)]

^1+d

. The approximation of the bathymetry map will be done in P (T

h

). The global shape functions in P(T

h

) are denoted by {ϕ

i

}

_i∈{1:I}

; the set {ϕ

i

}

_i∈{1:I}

is a basis of P (T

h

).

The partition of unity property on the reference shape functions implies that X

i∈{1:I}

ϕ

_i

(x) = 1 ∀x ∈ D.

(2.4)

Let D

i

be the support of ϕ

i

and |D

i

| be the measure of D

i

, i ∈ {1:I}. For any union of cells E ⊂ T

h

, we define I(E) := {j ∈ {1: I} | |D

j

∩ E| 6= 0} to be the set that contains the indices of all the shape functions whose support on E is of nonzero measure. We are going to regularly invoke I(K) and I(D

i

) and the partition of unity property P

i∈I(K)

ϕ

_i

(x) = 1 for all x ∈ K.

Let M be the consistent mass matrix with entries m

ij

:= R

D

ϕ

i

(x)ϕ

j

(x) dx, and let M

^L

be the diagonal lumped mass matrix with entries m

i

:= R

D

ϕ

i

(x) dx. The partition of unity property implies that m

i

= P

j∈I(Di)

m

ij

. One key assumption that we use in the rest of the chapter is that

(2.5) m

i

> 0 ∀i ∈ {1: I}.

The identities (2.4) are satisfied by all the standard finite elements and (2.5) is satisfied by many Lagrange elements and by the Bernstein–Bezier elements of any degree.

Upon denoting by k · k

_`2

the Euclidean norm in R

^d

, we introduce the following two quantities which will play an important role in the rest of the paper:

(2.6) c

ij

:=

Z

D

ϕ

i

∇ϕ

j

dx, n

ij

:= c

ij

kc

ij

k

_`2

, i, j ∈ {1: I}.

Note that (2.4) implies P

j∈{1:I}

c

_ij

= 0. Furthermore, if either ϕ

_i

or ϕ

_j

is zero on

∂D, then c

ij

= −c

ji

. In particular we have P

i∈{1:I}

c

ij

= 0 if ϕ

j

is zero on ∂D.

This property will be used to establish conservation.

Lemma 2.1. Let k ∈ C

¹

( R

^1+d

; R

^(1+d)×d

). Let u

h

= P

j∈{1:I}

U

j

ϕ

j

∈ P (T

h

).

Then P

j∈I(Di)

k(U

j

)·c

ij

is a second-order approximation of R

D

∇·(k(u

h

))ϕ

i

dx.

Proof. Since we have R

Di

∇·(k(u

_h

))ϕ

_i

dx = P

j∈{1:I}

k(U

_j

) R

Di

ϕ

_i

∇ϕ

_j

dx when k is linear, the quantity P

j∈I(Di)

k(U

j

)·c

ij

is a second-order approximation in space of R

D

∇·(k(u

h

))ϕ

i

dx, i.e., the error scales like O(h

²

)kc

ij

k

_`2

.

Definition 2.2 (centrosymmetry). The mesh T

h

is said to be centrosymmetric if the following conditions hold true: (i) For all i ∈ {1: I}, there is a permuta- tion σ

i

: I(D

i

) → I(D

i

) such that c

ij

= −c

_iσi(j)

; (ii) if the function D

i

3 x → P

j∈I(D_i)

α

_j

ϕ

_j

(x) ∈ R is linear over D

_i

then α

_i

=

¹₂

(α

_j

+ α

_σi(j)

) for all j ∈ I (D

_i

).

For instance, in the context of Lagrange elements, the centrosymmetric assump- tion holds if for any i ∈ {1: I} the set of the Lagrange nodes with indices in I (D

i

) can be partitioned into pairs that are symmetric with respect to the Lagrange node of index i. Although at some point in the paper we will invoke centrosymmetry of the mesh to establish formal consistency of some terms, we do not assume that the mesh is centrosymmetric in the rest of the paper.

2.3. Well-balancing properties. The concept of well-balancing originates in the seminal work of Bermudez and Vazquez [4] and Greenberg and Leroux [15]. The idea is that the scheme should at the very least preserve steady states at rest. Of course, it could be desirable to preserve general steady solutions, i.e., not necessarily at rest, but this is beyond the scope of the present paper. We refer the reader to Noelle, Xing, and Shu [26] where this question is addressed. Since at rest q = 0 the balance of momentum reduces to 0 = g∇(

¹₂

h

²

) + gh∇z = gh∇(h+z), one should have either h+ z is constant (so-called wet state) or h is zero (so-called dry state). Hence a well-balanced scheme in the context of the shallow water equation is one such that, at rest, dry states remain dry and h+z remains constant for wet states. This property is not easy to satisfy for approximation techniques that are second order and higher in space. We refer the reader to Bouchut [7] for a concise account and further references on well-balanced schemes. In this paper we are going to adapt to continuous finite elements a methodology proposed in Audusse and Bristeau [1], Audusse et al. [2]

known as the “hydrostatic reconstruction” technique.

Let z

_h

= P

I

i=1

Z

_i

ϕ

_i

∈ P (T

_h

) be the approximation of the bathymetry map.

Let h

h

= P

I

i=1

H

i

ϕ

i

∈ P(T

h

) be the approximation of the water height. Let q

h

= P

I

i=1

Q

i

ϕ

i

be the approximation of the flow rate. Let us now define the rest state.

Curiously, defining a rest state is not as trivial as it sounds. We are going to use two definitions. One of them makes use of the following quantity which is known in the literature as the hydrostatic reconstruction of the water height:

(2.7) H

^∗,j_i

:= max(0, H

_i

+ Z

_i

− max(Z

_i

, Z

_j

)) ∀i ∈ {1: I}, j ∈ I(D

i

).

To better understand this definition, assume that the water is at rest and consider for instance a dry node j in the neighborhood of a wet node i, i.e., j ∈ I(D

i

), see the left panel of Figure 1. In this case H

j

= 0 and Z

j

≥ H

i

+ Z

i

, which then implies H

^∗,j_i

= H

^∗,i_j

. Similarly if both i and j are dry states we have H

^∗,j_i

= H

^∗,i_j

, and if both i and j are wet states and are such that H

_j

+ Z

_j

= H

_i

+ Z

_i

we also have H

^∗,j_i

= H

^∗,i_j

. These observations motivate the following definition.

Definition 2.3 (rest at large). A numerical state (h

h

, q

h

, z

h

) is said to be at rest at large if the approximate momentum q

h

is zero, and if the approximate water height h

h

and the approximate bathymetry map z

h

satisfy the following property, for all i ∈ {1: I}: H

^∗,j_i

= H

^∗,i_j

for all j ∈ I (D

_i

).

Definition 2.4 (exact rest). A numerical state (h

h

, q

h

, z

h

) is said to be at exact rest (or exactly at rest) if q

h

is zero, and if the approximate water height h

h

and the approximate bathymetry map z

h

satisfy the following alternative, for all i ∈ {1: I}:

for all j ∈ I (D

i

), either H

j

= H

i

= 0 or H

j

+ Z

j

= H

i

+ Z

i

.

(a) (b) (c)

Fig. 1.Configuration(a)is not an exact rest state according to Definition2.4whereas config- uration(b)is. Both states are at rest at large. Panel (c)shows a typical steady state at rest with wet and dry areas.

The existence of an exact rest state is a compatibility condition between the mesh and the initial data. This compatibility condition is not satisfied by the configuration depicted in the left panel of Figure 1 whereas it is satisfied by the configuration in the center panel. Exact rest implies rest at large. Note in passing that the zone where h + z is constant may not be connected; that is to say, it is possible to have different free surface heights in disconnected wet zones as shown in the right panel of Figure 1.

Definition 2.5 (well-balancing at large). (i) A function K : P (T

h

) → R

^I

×( R

^I

)

^d

is said to be a well-balanced flux approximation at large if K(u

h

) = 0 when u

h

is a rest state at large according to Definition 2.3. (ii) A mapping S : P (T

h

) → P (T

h

) is a well-balanced scheme at large if S(u

h

) = u

h

when u

h

is a rest state at large.

Definition 2.6 (exact well-balancing). (i) A function K : P (T

h

) → R

^I

×( R

^I

)

^d

is said to be an exactly well-balanced flux approximation if K(u

h

) = 0 when u

h

is an exact rest state according to Definition 2.4. (ii) A mapping S : P (T

h

) → P (T

h

) is an exactly well-balanced scheme if S(u

ⁿ_h

) = u

ⁿ_h

when u

ⁿ_h

is an exact rest state.

Definition 2.7 (conservation). We say that u

ⁿ_h

→ u

ⁿ⁺¹_h

is a conservative fi- nite element approximation of (2.1) if P

i∈{1:I}

m

_i

H

ⁿ_i

= P

i∈{1:I}

m

_i

H

ⁿ⁺¹_i

and if P

i∈{1:I}

m

_i

Q

ⁿ_i

= P

i∈{1:I}

m

_i

Q

ⁿ⁺¹_i

when the topography map is constant.

3. First order scheme. We describe in this section a time and space approx- imation of (2.2). The scheme is well-balanced at large but approximates the flux to first order in space only. This scheme satisfies local invariant domain properties and local discrete entropy inequalities when the bottom is flat. It is an adaptation of the method presented in Audusse et al. [2] to the continuous finite element setting developed in Guermond and Popov [16]. To the best of our knowledge, this is the first result of this type for continuous finite elements.

3.1. Flux approximation. Just like in [2, (2.13)], the key is to consider the hydrostatic reconstruction (2.7) and to observe that P

j∈I(D_i) 1

2

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

)c

_ij

is a well-balanced first-order approximation of the flux R

D_i

(∇(

¹₂

h

²

) + h∇z)ϕ

i

dx.

Lemma 3.1 (consistency/well-balancing). (i) Assume that {b θ

_n

}

_n∈{1:nsh}

con- sists of Lagrange or Bernstein functions. Then P

j∈I(Di) 1

2

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

)c

ij

is a first-order approximation of the flux R

Di

(∇(

¹₂

h

²

) + h∇z)ϕ

_i

dx. (ii) The mapping u

h

→ (0, P

j∈I(Di) 1

2

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

)c

ij

)

_i∈{1:I}

is well-balanced at large.

Proof. (i) Let us fix i ∈ {1: I}. We slightly abuse the notation by using h to

denote the mesh size. For the consistency analysis we assume that the water height

and the bathymetry map are smooth and the water height is nonnegative. More

precisely, we assume that there is C

z

such that for all i ∈ {1: I}, |Z

i

− Z

j

| ≤ C

z

h for all j ∈ I(D

i

).

Assume first that Z

j

≥ Z

i

. We immediately get H

^∗,i_j

= H

j

. If, in addition, H

i

≥ C

z

h, then H

^∗,j_i

= max(0, H

i

+ (Z

i

− Z

j

)) = H

i

+ (Z

i

− Z

j

), and we have

1

2

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

) =

¹₂

H

²_j

−

¹₂

(H

i

+ (Z

i

− Z

j

))

²

=

¹₂

H

²_j

−

¹₂

H

²_i

+ H

i

(Z

j

− Z

i

) + O(h

²

).

Similarly, if H

_i

≤ C

_z

h, then H

^∗,j_i

= O(h) and we again have

¹₂

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

) =

1

2

H

²_j

−

¹₂

H

²_i

+ H

i

(Z

j

− Z

i

) + O(h

²

). On the other hand, if Z

i

≤ Z

j

, we obtain

1

2

((H

^∗,i_j

)

²

−(H

^∗,j_i

)

²

) =

¹₂

H

²_j

−

¹₂

H

²_i

+H

_j

(Z

_j

−Z

_i

) +O(h

²

). But since H

_j

= H

_i

+O(h) (we are using continuous finite elements and the water height is assumed to be smooth), we also have

¹₂

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

) =

¹₂

H

²_j

−

¹₂

H

²_i

+ H

_i

(Z

_j

− Z

_i

) + O(h

²

) in this case.

Using Lemma 2.1 we infer that P

j∈I(D_i)

(

¹₂

H

²_j

−

¹₂

H

²_i

)c

_ij

is a second-order ap- proximation of R

D

(∇(

¹₂

h

²

))ϕ

i

dx. Similarly, P

j∈I(Di)

(H

i

(Z

j

− Z

i

))c

ij

is a second- order approximation of H

_i

R

D

(∇z)ϕ

_i

dx. If z is linear over D

_i

(which is a sufficient assumption for the consistency analysis), then H

_i

R

D

(∇z)ϕ

_i

dx = ∇z

_|Di

H

_i

R

D

ϕ

_i

dx.

Since H

_i

R

D

ϕ

_i

dx can be shown to be a second-order approximation of R

Di

hϕ

_i

dx (at least for Lagrange and Bernstein basis functions), we conclude that P

j∈I(Di)

(H

i

(Z

j

− Z

i

))c

ij

is a second-order approximation of R

D

(h∇z)ϕ

i

dx. Com- bining these observations with the above argument and upon observing that kc

ij

k

`²

O(h

²

) = m

_i

O(h), we conclude that P

j∈I(Di) 1

2

((H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

)c

_ij

is a first- order approximation of R

D

(∇(

¹₂

h

²

) + h∇z)ϕ

i

dx.

(ii) Let us prove the well-balancing at large. Assuming that u

h

is a rest state at large, according to Definition 2.3 we have H

^∗,i_j

= H

^∗,j_i

, hence (H

^∗,i_j

)

²

− (H

^∗,j_i

)

²

= 0.

The conclusion follows immediately.

Let us introduce the gas dynamics flux g(u) := (q,

¹_h

q ⊗ q)

^T

. We now need to approximate R

D_i

g(u)ϕ

i

dx. Since we have seen above that using H

^∗

is a good idea to guarantee well-balancing at large, one could imagine working with the pair (H

^∗,j_i

, Q

i

)

^T

. The problem with this choice is that if it happens that H

^∗,j_i

is zero (because H

i

+ Z

i

≤ max(Z

i

, Z

j

)), there is no reason for the approximate flow rate Q

_i

to be zero; hence the quantity Q

_i

/H

^∗,j_i

which approximates the velocity could be unbounded. To avoid this problem, we proceed as in [2] by working with the quantities (3.1) Q

^∗,j_i

:= Q

_i

H

^∗,j_i

H

i

, U

^∗,j_i

:=

H

^∗,j_i

, Q

^∗,j_i

^T

with the convention that Q

^∗,j_i

:= 0 if H

_i

= 0. Note that we have kQ

^∗,j_i

k

_`2

≤ kQ

_i

k

_`2

since 0 ≤ H

^∗,j_i

≤ H

i

by definition. We now face the question of constructing a consistent approximation of R

D_i

g(u)ϕ

i

dx using the state variable U

^∗,j_i

. To simplify the notation let us introduce the approximate velocity v

_h

= P

i∈{1:I}

V

_i

ϕ

_i

with

(3.2) V

i

:= Q

i

H

_i

, i ∈ {1: I}.

Definition 3.2 (shoreline). We say that a degree of freedom i is away from the shoreline if either H

j

= 0 for all j ∈ I (D

i

) or min(H

j

, H

i

) > |Z

i

−Z

j

| for all j ∈ I(D

i

).

Note that if the bottom topography is smooth, i.e., there is C

z

such that for all

i ∈ {1: I}, |Z

i

− Z

j

| ≤ C

z

h, then any degree of freedom i such that H

j

≥ C

z

h for

all j ∈ I(D

i

), is away from the shoreline according to the above definition. Roughly

speaking, a degree of freedom i is said to be away from the shoreline if either all the

degrees of freedom around i are dry or the water depth around i is at least C

z

h if the bottom topography is smooth (h being the mesh size).

Lemma 3.3. The quantity P

j∈I(Di)

(g(U

^∗,i_j

) +g(U

^∗,j_i

))·c

ij

is a first-order approx- imation of R

D_i

∇·g(u)ϕ

i

dx away from the shoreline if the mesh is centrosymmetric.

Proof. Let i ∈ {1:I} be a degree of freedom away from the shoreline. The ap- proximation of the flux is P

j∈I(Di)

(V

j

H

^∗,i_j

+ V

i

H

^∗,j_i

))·c

ij

for the mass conservation equation and P

j∈I(Di)

((V

j

⊗ V

j

)H

^∗,i_j

+ (V

i

⊗ V

i

)H

^∗,j_i

))·c

ij

for the flow rate conser- vation. Let us start with the mass conservation equation. We proceed as in the proof of Lemma 3.1 and again assume that the water height and the bathymetry map are smooth and the water height is nonnegative. Since the mesh is centrosymmetric by hypothesis, we can assume without loss of generality that Z

_j

≥ Z

_i

≥ Z

_σi(j)

. Then H

^∗,i_j

= H

_j

and since i is away from the shoreline we have either H

^∗,j_i

= H

_i

+ Z

_i

− Z

_j

if H

i

6= 0, or H

^∗,j_i

= 0 if H

i

= 0. Similarly, H

^∗,σ_i ^i(j)

= H

i

and since i is away from the shoreline we have either H

^∗,i_σ

i(j)

= H

_σi(j)

+ Z

_σi(j)

− Z

_i

if H

_σi(j)

6= 0, or H

^∗,i_σ

i(j)

= 0 if H

_σi(j)

= 0. Hence, if i is a wet state (and all the states in I(D

i

) are wet since i is away from the shoreline), we have

V

j

H

^∗,i_j

+ V

i

H

^∗,j_i

·c

ij

+

V

_σi(j)

H

^∗,i_σ

i(j)

+ V

i

H

^∗,σ_i ^i(j)

·c

_iσi(j)

= V

_j

H

_j

+ V

_i

(H

_i

+ Z

_i

− Z

_j

) − (V

_σi(j)

(H

_σi(j)

+ Z

_σi(j)

− Z

_i

) + V

_i

H

_i

)

·c

ij

= (V

j

H

j

− V

i

H

i

)·c

ij

+ (V

_σi(j)

H

_σi(j)

− V

i

H

i

)·c

_iσi(j)

+ V

_i

(Z

_i

− Z

_j

)·c

_ij

+ V

_σi(j)

(Z

_σi(j)

− Z

_i

)·c

_iσi(j)

,

where we have used the centrosymmetry property c

ij

= −c

_iσi(j)

. If i is a dry state (recall that j and σ

i

(j) are also dry states since i is away from the shoreline) then

V

j

H

^∗,i_j

+ V

i

H

^∗,j_i

·c

ij

+

V

_σi(j)

H

^∗,i_σ

i(j)

+ V

i

H

^∗,σ_i ^i(j)

·c

_iσi(j)

= (V

_j

H

_j

− V

_i

H

_i

)·c

_ij

+ (V

_σi(j)

H

_σi(j)

− V

_i

H

_i

)·c

_iσi(j)

. Since according to Lemma 2.1, P

j∈I(Di)

(V

j

H

j

− V

i

H

i

)·c

ij

= P

j∈I(Di)

V

j

H

j

·c

ij

is a second-order approximation of R

D

∇·(v

h

h

h

)ϕ

i

dx, we have to show that the contribution of the extra term V

i

(Z

i

− Z

j

)·c

ij

− V

σ_i(j)

(Z

σ_i(j)

− Z

i

)·c

ij

that arises when i is a wet state is small. Assuming that the velocity is smooth, we have V

_σi(j)

= V

_i

+ O(h), which shows that V

_i

(Z

_i

− Z

_j

)·c

ij

− V

_σi(j)

(Z

_σi(j)

− Z

_i

)·c

ij

= V

_i

(2Z

_i

− Z

_j

− Z

_σi(j)

)·c

_ij

+ kc

_ij

k

_`2

O(h

²

). The centrosymmetry assumption implies that 2Z

_i

− Z

_j

− Z

_σi(j)

= O(h

²

) if the bathymetry map is smooth. In conclusion P

j∈I(Di)

(V

j

H

^∗,i_j

+ V

i

H

^∗,j_i

)·c

ij

= P

j∈I(Di)

V

j

H

j

·c

ij

+ m

i

O(h) away from the shore- line. Using the same argument one proves that

X

j∈I(Di)

((V

j

⊗ V

j

)H

^∗,i_j

+ (V

i

⊗ V

i

)H

^∗,j_i

))·c

ij

= X

j∈I(Di)

(V

j

⊗ V

j

)H

j

+ m

i

O(h).

This concludes the proof.

Remark 3.4 (hydrostatic reconstruction). The lack of consistency of the hydro-

static reconstruction at the shoreline or in the presence of large gradients in the

topography map has been identified in Delestre et al. [10, Prop. 2.1]. Various alter-

natives to the hydrostatic reconstruction have since been proposed like in Berthon

and Foucher [5], Bryson et al. [9], Duran, Liang, and Marche [12], where the authors

propose to work with the free surface elevation instead of the water height.

3.2. Full time and space approximation. Let u

⁰_h

= P

I

i=1

U

⁰_i

ϕ

i

∈ P (T

h

) be a reasonable approximation of u

0

. Let n ∈ N , τ be the time step, t

n

be the current time, and let us set t

n+1

= t

n

+ τ. Let u

ⁿ_h

= P

I

i=1

U

ⁿ_i

ϕ

i

∈ P (T

h

) be the space approximation of u at time t

n

. Upon denoting H

^∗,j,n_i

:= max(0, H

ⁿ_i

+Z

i

−max(Z

i

, Z

j

)), we propose to estimate U

ⁿ⁺¹_i

as follows:

(3.3) m

i

U

ⁿ⁺¹_i

− U

ⁿ_i

τ + X

j∈I(Di)

(g(U

^∗,i,n_j

) + g(U

^∗,j,n_i

))·c

ij

+

0

1

2

g (H

^∗,i,n_j

)

²

− (H

^∗,j,n_i

)

²

c

ij

− X

i6=j∈I(Di)

d

ⁿ_ij

(U

^∗,i,n_j

− U

^∗,j,n_i

) = 0, where the artificial viscosity coefficient d

ⁿ_ij

is defined by

d

ⁿ_ij

:= max

d

^f,n_ij

, d

^f,n_ji

, (3.4)

d

^f,n_ij

:= max λ

^f_max

n

ij

, U

ⁿ_i

, U

^∗,i,n_j

, λ

^f_max

n

ij

, U

ⁿ_i

, U

^∗,j,n_i

kc

ij

k

_`2

, (3.5)

and λ

^f_max

(n, U

L

, U

R

) is the maximum wave speed in the Riemann problem:

(3.6) ∂

_t

u + ∂

_x

(f (u)·n) = 0, u(x, 0) = (1 − H(x))U

_L

+ H (x)U

_R

,

where H(x) is the Heaviside function. Note that d

ⁿ_ij

≥ 0 and d

ⁿ_ij

= d

ⁿ_ji

for all j 6= i in I(D

i

). For convenience we denote d

ⁿ_ii

:= − P

i6=j∈I(Di)

d

ⁿ_ij

. Therefore we have P

j∈I(Di)

d

ⁿ_ij

= P

j∈I(Di)

d

ⁿ_ji

= 0; this property will be used in the rest of the paper.

3.3. Reduction to the one-dimensional Riemann problem. For complete- ness, we show how the estimation of λ

^f_max

(n, U

L

, U

R

) can be reduced to estimating the maximum wave speed in a one-dimensional Riemann problem independent of n.

Similarly to [16], we make a change of basis and introduce t

1

, . . . , t

_d−1

∈ R

^d

so that {n, t

1

, . . . , t

_d−1

} is an orthonormal basis of R

^d

. With respect to this basis we have that q = (q, q

^⊥

), where q := q·n and q

^⊥

:= (q·t

1

, . . . , q·t

_d−1

)

^T

. Then, with the no- tation v = q/h, the Riemann problem (3.6) can be rewritten in the new orthonormal basis as follows:

(3.7) ∂

t

u + ∂

x

(n·f (u)) = 0, u =



 h q q

^⊥



 , f (u)·n =



 q vq +

^g₂

h

²

vq

^⊥





with data U

L

= (h

L

, q

L

, q

^⊥_L

)

^T

, U

R

= (h

R

, q

R

, q

_R^⊥

)

^T

. The solution to (3.7) is henceforth denoted u(n, U

L

, U

R

)(x, t). Following [16], we introduce the following definition.

Definition 3.5 (invariant set). A convex set A ⊂ A is said to be invariant for the flat bottom system, i.e., (2.1) with b = 0, if for any admissible pair (U

L

, U

R

) ∈ A×A and any unit vector n ∈ R

^d

, we have u(n, U

L

, U

R

)(x, t) ∈ A for a.e. x ∈ R , t > 0.

Let u(t, n, U

_L

, U

_R

) := R

¹₂

−¹₂

u(n, U

_L

, U

_R

)(x, t) dx. Then, the following result is a consequence of λ

^f_max

(n, U

_L

, U

_R

) being finite; see [16, Lem. 2.1].

Lemma 3.6 (invariant set and average). (i) Let A ⊂ A be an invariant set for the flat bottom system. If (U

L

, U

R

) ∈ A, then u(t, n, U

L

, U

R

) ∈ A. (ii) Assume that 2t λ

max

(n, U

L

, U

R

) ≤ 1, then u(t, n, U

L

, U

R

) =

¹₂

(U

L

+ U

R

) − t(f (U

R

) − f (U

L

))·n.

This lemma is the key motivation for the definition of the viscosity coefficients

d

^f,n_ij

in (3.5) (see [16, section 3.3] for more details).

The maximum wave speed in the Riemann problem (3.7) is determined by the one-dimensional shallow water system for the component (h, q)

^T

because the last component is just passively transported and does not influence the first two equations of the system. That is to say (3.7) reduces to solving the Riemann problem

(3.8) ∂

t

(h, q)

^T

+ ∂

x

(f

1D

(h, q)) = 0

with data u

L

:= (h

L

, q

L

), u

R

:= (h

R

, q

R

) and flux f

1D

(h, q) := (q, vq +

^g₂

h

²

)

^T

. This establishes the following result which will be useful to estimate d

^f,n_ij

in (3.5).

Proposition 3.7 (maximum wave speed). Let λ

^f_max

(n, U

_L

, U

_R

), λ

^f_max^1D

(u

_L

, u

_R

) be the maximum wave speed in the Riemann problems (3.7) and (3.8), respectively.

Then λ

^f_max

(n, U

_L

, U

_R

) = λ

^f_max^1D

(u

_L

, u

_R

).

In order to estimate λ

^f_max^1D

(u

_L

, u

_R

) from above, we introduce

λ

⁻₁

(h

_∗

) := v

L

− p

gh

L

1 +

h

_∗

− h

L

2h

L

+

!

¹₂

1 +

h

_∗

− h

L

h

L

+

!

¹₂

, (3.9)

λ

⁺₂

(h

_∗

) := v

R

+ p

gh

R

1 +

h

∗

− h

R

2h

R

+

!

¹₂

1 +

h

∗

− h

R

h

R

+

!

¹₂

. (3.10)

The following result is proved in Guermond and Popov [18]:

Lemma 3.8. Let h

_min

= min(h

_L

, h

_R

), h

_max

= max(h

_L

, h

_R

), x

₀

= (2 √

2 − 1)

²

, and

h

_∗

:=



 

 



 

 



(vL−vR+2√ ghL+2√

ghR)²₊

16g

if case 1,

− √

2h

min

+ r

3h

min

+ 2 √

2h

min

h

max

+ q

2

g

(v

L

− v

R

) √ h

min

²

if case 2,

√ h

min

h

max

1 +

√2(v_L−v_R)

√gh_min+√ gh_max

if case 3, where case 1 is 0 ≤ f (x

₀

h

_min

), case 2 is f (x

₀

h

_min

) < 0 ≤ f (x

₀

h

_max

), and case 3 is f (x

₀

h

_max

) < 0. Then λ

^f_max

(n, U

_L

, U

_R

) = λ

^f_max^1D

(u

_L

, u

_R

) ≤ max(|λ

⁻₁

(h

_∗

)|, |λ

⁺₂

(h

_∗

)|).

3.4. Stability properties. We collect in this section some remarkable stability properties of the scheme defined by (3.3)–(3.5).

Proposition 3.9 (well-balancing/conservation). The scheme defined in (3.3) is well-balanced at large, and it is conservative in the sense of Definition 2.7.

Proof. Let u

ⁿ_h

be a rest state at large, then H

^∗,i,n_j

= H

^∗,j,n_i

for all i ∈ {1: I}

and all j ∈ I(D

i

); this identity implies well-balancing at large. Let us now establish conservation. Since c

_ij

= −c

ji

and d

ⁿ_ij

= d

ⁿ_ji

we have

X

i∈{1:I}

X

j∈I(Di)

c

ji

α

ij

= 0, X

i∈{1:I}

X

j∈I(Di)

d

ⁿ_ji

β

ij

= 0

for any symmetric field α

ij

= α

ji

and any skew-symmetric field β

ij

= −β

ij

. Hence, we only have to deal with the nonconservative flux in (3.3),

¹₂

g((H

^∗,i,n_j

)

²

− (H

^∗,j,n_i

)

²

)c

ij

. This quantity is zero for a constant topography map. This concludes the proof.

Since the shallow water system makes sense only for nonnegative water heights,

and the water discharge should be zero in dry states, we are led to consider the

following definition for the admissibility of shallow water states.

Definition 3.10 (admissible water states). A shallow water state U = (H, Q)

^T

is admissible if H ≥ 0 and Q = 0 if H = 0. The set of admissible states is denoted A.

Note that a convex combination of admissible states is always an admissible state.

Proposition 3.11 (invariant domain). Let u

ⁿ⁺¹_h

be given by (3.3)–(3.5), n ≥ 0.

Let ∈ {1: I}. Assume that 1 + 4

_m^τ

i

d

ⁿ_ii

≥ 0. Let A

ⁿ_i

be an invariant set of the shallow water equation that contains {U

ⁿ_j

}

_j∈I(Di)

. Then the following properties hold true:

(i) If the bathymetry map is constant then U

ⁿ⁺¹_i

∈ A

ⁿ_i

. (ii) If the bathymetry is not constant, let

∆Z

_iⁿ

:= τ m

i

X

i6=j∈I(D_i)

g((H

ⁿ_i

)

²

− (H

^∗,j,n_i

)

²

)c

_ij

and ∆U

^∗,n_i

:=

_m^2τ

i

P

i6=j∈I(Di)

d

ⁿ_ij

(1 −

^H∗,j,ni_Hn i

)U

ⁿ_i

then U

ⁿ⁺¹_i

∈ conv(A

ⁿ_i

, 0) + (0, ∆Z

_iⁿ

)

^T

+

∆U

^∗,n_i

; in particular the scheme preserves the nonnegativity of the water height.

(iii) If the states {U

ⁿ_i

} are admissible then the states {U

ⁿ⁺¹_i

} are also admissible.

Proof. Recalling that f (u) = g(u)+(0,

¹₂

gh

²

I

d

)

^T

, then (3.3) can also be rewritten m

i

τ U

ⁿ⁺¹_i

− U

ⁿ_i

+ X

j∈I(Di)

f U

^∗,i,n_j

·c

_ij

− d

ⁿ_ij

U

^∗,i,n_j

+ f U

^∗,j,n_i

·c

_ij

− d

ⁿ_ij

U

^∗,j,n_i

+ X

j∈I(Di)

0, −g

H

^∗,j,n_i

2

c

ij

^T

+ (d

ⁿ_ij

+ d

ⁿ_ij

)U

^∗,j,n_i

= 0.

Using conservation, i.e., c

ii

= − P

i6=j∈I(Di)

c

ij

, this equation can be recast into m

_i

τ U

ⁿ⁺¹_i

− U

ⁿ_i

= X

i6=j∈I(Di)

− f

U

^∗,i,n_j

− f (U

ⁿ_i

)

·c

ij

+ d

ⁿ_ij

U

^∗,i,n_j

+ U

ⁿ_i

+ X

i6=j∈I(Di)

− f

U

^∗,j,n_i

− f (U

ⁿ_i

)

·c

ij

+ d

ⁿ_ij

U

^∗,j,n_i

+ U

ⁿ_i

+ X

i6=j∈I(Di)

0, g

(H

ⁿ_i

)

²

−

H

^∗,j,n_i

²

c

ij

T

− d

ⁿ_ij

+ d

ⁿ_ij

U

^∗,j,n_i

+ U

ⁿ_i

.

Upon introducing the vectors U

ⁿ_ij

∈ R

^1+d

, W

ⁿ_ij

∈ R

^1+d

, and ∆Z

_iⁿ

∈ R

^d

defined by

U

ⁿ_ij

:= − kc

_ij

k

_`2

2d

ⁿ_ij

f U

^∗,i,n_j

− f (U

ⁿ_i

)

·n

_ij

+ 1 2

U

^∗,i,n_j

+ U

ⁿ_i

,

W

ⁿ_ij

:= − kc

ij

k

_`2

2d

ⁿ_ij

f U

^∗,j,n_i

− f (U

ⁿ_i

)

·n

ij

+ 1 2

U

^∗,j,n_i

+ U

ⁿ_i

,

∆Z

_iⁿ

:= X

i6=j∈I(Di)

g

(H

ⁿ_i

)

²

−

H

^∗,j,n_i

2

c

_ij

,

we finally obtain U

ⁿ⁺¹_i

=



1 − X

i6=j∈I(Di)

4τ m

i

d

ⁿ_ij



 U

ⁿ_i

+ X

i6=j∈I(Di)

2τ m

i

d

ⁿ_ij

(U

ⁿ_ij

+ W

ⁿ_ij

)

+ τ

m

_i

(0, ∆Z

_iⁿ

)

^T

+ 2τ m

_i

X

i6=j∈I(Di)

d

ⁿ_ij

1 − H

^∗,j,n_i

H

ⁿ_i

! U

ⁿ_i

.

Upon introducing the fake time t =

^kc_2d^ijn^k`2 ij

and observing that the definition of d

ⁿ_ij

implies that 2tλ

^f_max

(n

ij

, U

ⁿ_i

, U

^∗,i,n_j

) ≤ 1 and 2tλ

^f_max

(n

ij

, U

ⁿ_i

, U

^∗,j,n_i

) ≤ 1, we infer from Lemma 3.6 that U

ⁿ_ij

∈ conv

_j∈I(Di)

(U

^∗,i,n_j

) and W

ⁿ_ij

∈ conv

_j∈I(Di)

(U

^∗,j,n_i

); hence,

Uⁿ_ij+Wⁿ_ij

2

∈ conv

_j∈I(Di)

(U

^∗,i,n_j

, U

^∗,j,n_i

). In conclusion, under the CFL condition 1 + 4

_m^τ

i

d

ⁿ_ii

≥ 0, the state U e

ⁿ⁺¹_i

:= (1 +

_m^4τ

i

d

ⁿ_ii

)U

ⁿ_i

+ P

i6=j∈I(Di) 2τ

m_i

d

ⁿ_ij

(U

ⁿ_ij

+ W

ⁿ_ij

) belongs to conv

_j∈I(Di)

(U

^∗,i,n_j

, U

^∗,j,n_i

). If the bathymetry map is flat then H

ⁿ_i

= H

^∗,j,n_i

and we obtain U

ⁿ⁺¹_i

= U e

ⁿ⁺¹_i

∈ conv

_j∈I(Di)

(U

ⁿ_j

) ⊂ A

ⁿ_i

and this proves (i). If the bathymetry is not flat, then U

^∗,i,n_j

is in the convex hull of U

ⁿ_j

and 0 for all j ∈ I(D

i

) and U

^∗,j,n_i

is in the convex hull of U

ⁿ_i

and 0 for all j ∈ I(D

i

); this proves that U e

ⁿ⁺¹_i

∈ conv(A

ⁿ_i

, 0).

Hence, if the bathymetry is not flat we get U

ⁿ⁺¹_i

∈ conv(A

ⁿ_i

, 0) + (0, ∆Z

_iⁿ

)

^T

+ ∆U

^∗,n_i

as announced. The water height in ∆U

^∗,n_i

is

_m^2τ

i

P

i6=j∈I(Di)

d

ⁿ_ij

(H

ⁿ_i

− H

^∗,j,n_i

) ≥ 0.

Since all the states in A

ⁿ_i

have nonnegative water height, we conclude that H

ⁿ⁺¹_i

≥ 0 and this proves (ii). Finally, fix n ≥ 0 and assume that all states {U

ⁿ_j

} are admissible in the sense of Definition 3.10. If H

ⁿ_i

> 0 then we have that

H

ⁿ⁺¹_i

≥



1 − X

i6=j∈I(Di)

4τ m

i

d

ⁿ_ij



 H

ⁿ_i

> 0,

and this proves that U

ⁿ⁺¹_i

is admissible. In the remaining case H

ⁿ_i

= 0, we have that H

^∗,j,n_i

= 0 for all j ∈ I(D

i

) and ∆Z

_iⁿ

= 0. Hence U

ⁿ⁺¹_j

= U e

ⁿ⁺¹_i

and using that U e

ⁿ⁺¹_i

is a convex combination of admissible states we conclude that the state U

ⁿ⁺¹_i

is admissible and this proves (iii).

We finish with a discrete inequality which reduces to a standard discrete entropy inequality when the bottom topography is flat. The proof is omitted for brevity.

Proposition 3.12. Let u

ⁿ⁺¹_h

be given by (3.3)–(3.5). Assume the CFL condition 1 + 4

_m^τ

i

d

ⁿ_ii

≥ 0. Then for any flat bed shallow water entropy pair (η, G), we have the following discrete entropy inequality:

(3.11) m

i

τ η U

ⁿ⁺¹_i

− η (U

ⁿ_i

)

+ X

i6=j∈I(Di)

G

U

^∗,i,n_j

+ G

U

^∗,j,n_i

·c

ij

≤ X

i6=j∈I(Di)

d

ⁿ_ij

η

U

^∗,i,n_j

+ η

U

^∗,j,n_i

− 2η (U

ⁿ_i

)

+



(0, ∆Z

_iⁿ

)

^T

+ X

i6=j∈I(Di)

2d

ⁿ_ij

1 − H

^∗,j,n_i

H

ⁿ_i

! U

ⁿ_i



 ·∇η(U

ⁿ⁺¹_i

).

Remark 3.13 (literature). We refer the reader to Bouchut and Frid [8, section 2]

for an alternative point of view to derive the invariant domain property and entropy inequality obtained above.

4. Second-order extension. In this section we propose a scheme that is second- order accurate in space, is exactly well-balanced, and is positivity preserving.

4.1. Flux approximation. We start by constructing a well-balanced second- order approximation of the quantity R

Di

(∇(

¹₂

h

²

) + h∇z)ϕ

i

dx.

Lemma 4.1 (consistency/well-balancing). (i) Assume that {b θ

n

}

_n∈{1:nsh}

con- sists of Lagrange or Bernstein basis functions. The expression P

j∈I(D_i)

H

_i

(H

_j

+Z

_j

)c

_ij

is a second-order approximation of R

D

(∇(

¹₂

h

²

) + h∇z)ϕ

i

dx. (ii) The mapping u

h

→ (0, P

j∈I(Di)

H

i

(H

j

+ Z

j

)c

ij

)

_i∈{1:I}

is an exactly well-balanced flux.

Proof. (i) If h + z is linear over K ∈ T

h

then R

K

h∇(h + z)ϕ

_i

dx =

∇(h + z)|

K

R

K

hϕ

i

dx and the approximation R

K

hϕ

i

dx ≈ H

i1

d

|K| is second-order accurate, at least for Lagrange and Bernstein basis functions. Hence, upon notic- ing that P

K⊂Di

∇(h + z)

_|K¹_d

|K| = R

D_i

∇(h + z)ϕ

i

dx = P

j∈I(Di)

(H

j

+ Z

j

)c

ij

, the expression R

D

h∇(h + z)ϕ

_i

dx ≈ P

j∈I(D_i)

H

_i

(H

_j

+ Z

_j

)c

_ij

is formally second-order accurate.

(ii) Let us now prove well-balancing. Let us assume exact rest. Let us fix i ∈ {1: I}. Notice that owing to the partition of unity property we have P

j∈I(Di)

c

_ij

= 0;

hence P

j∈I(Di)

H

i

(H

j

+ Z

j

)c

ij

= P

j∈I(Di)

H

i

(H

j

+ Z

j

− H

i

− Z

i

)c

ij

. Consider j ∈ I(D

i

). According to our definition of the exact rest state (see Definition 2.4), either H

i

= 0 and H

j

= 0, or H

j

+ Z

j

− H

i

− Z

i

= 0; whence the conclusion.

Let us introduce the gas dynamics flux g(u) := (q,

¹_h

q ⊗ q)

^T

; then upon invoking Lemma 2.1, P

j∈I(Di)

g(U

j

)·c

ij

is a second-order approximation of R

D_i

∇·(g(u))ϕ

i

dx.

4.2. Full time and space approximation. Let u

⁰_h

= P

I

i=1

U

⁰_i

ϕ

i

∈ P (T

h

) be a reasonable approximation of u

0

. Let n ∈ N , τ be the time step, t

n

be the current time, and t

n+1

:= t

n

+ τ. Let u

ⁿ_h

= P

I

i=1

U

ⁿ_i

ϕ

i

∈ P (T

h

) be the space approximation of u at time t

_n

and let u

ⁿ⁺¹_h

:= P

I

i=1

U

ⁿ⁺¹_i

ϕ

_i

. We estimate U

ⁿ⁺¹_i

as follows:

m

i

τ U

ⁿ⁺¹_i

− U

ⁿ_i

= X

j∈I(Di)

− g U

ⁿ_j

·c

ij

− 0, gH

ⁿ_i

H

ⁿ_j

+ Z

j

c

ij

^T

+ X

i6=j∈I(Di)

d

ⁿ_ij

U

^∗,i,n_j

− U

^∗,j,n_i

+ µ

ⁿ_ij

U

ⁿ_j

− U

^∗,i,n_j

−

U

ⁿ_i

− U

^∗,j,n_i

(4.1)

µ

ⁿ_ij

:= max((V

_i

·n

_ij

)

₋

, (V

_j

·n

_ij

)

₊

)kc

_ij

k

_`2

, d

ⁿ_ij

≥ µ

ⁿ_ij

, i 6= j.

(4.2)

Here we use the notation a

₊

:= max(a, 0) and a

₋

= − min(a, 0). In the above scheme d

ⁿ_ij

= d

ⁿ_ji

can be any nonnegative number larger than µ

ⁿ_ij

when i 6= j. One could just take d

ⁿ_ij

= µ

ⁿ_ij

, but a more robust choice consists of using d

ⁿ_ij

= max(d

^f,n_ij

, d

^f,n_ji

);

note that in this case the local maximum wave speed formulas (3.9) and (3.10) used with u

L

:= (H

ⁿ_i

, Q

ⁿ_i

·n

ij

) and u

R

= (H

ⁿ_j

, Q

ⁿ_i

·n

ij

) imply that d

ⁿ_ij

≥ µ

ⁿ_ij

. Notice that µ

ⁿ_ij

= µ

ⁿ_ji

because n

ij

= −n

ji

owing to the assumed boundary condition. We adopt again the convention d

ⁿ_ii

:= − P

i6=j∈I(Di)

d

ⁿ_ij

.

Proposition 4.2. The scheme (4.1)–(4.2) is exactly well-balanced and conserva- tive. It is positivity preserving provided 1 + 2d

ⁿ_iim^τ

i

≥ 0 for all i ∈ {1: I}.